Properties

Label 441.2.h.b
Level 441
Weight 2
Character orbit 441.h
Analytic conductor 3.521
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{3} ) q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{3} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{6} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{8} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{3} ) q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{3} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{6} + ( -1 - \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{8} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{9} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{10} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} + ( -1 - 3 \beta_{1} + \beta_{5} ) q^{12} + ( 1 + \beta_{4} ) q^{13} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{15} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{16} + ( -1 + 2 \beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{17} + ( 4 - \beta_{1} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{18} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{19} + ( 1 - 3 \beta_{1} + \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{20} + ( 5 - 2 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} ) q^{22} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{23} + ( 1 + 4 \beta_{1} + \beta_{2} - 2 \beta_{5} ) q^{24} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{26} + ( -2 + 2 \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{27} + ( -\beta_{1} + \beta_{5} ) q^{29} + ( 5 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{30} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{31} + ( -3 - \beta_{2} - \beta_{3} ) q^{32} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{33} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{34} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{36} + ( -3 \beta_{2} + 3 \beta_{5} ) q^{37} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{38} + ( -1 + \beta_{5} ) q^{39} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 5 \beta_{4} ) q^{40} + ( -7 + \beta_{2} - 7 \beta_{4} - \beta_{5} ) q^{41} + ( -1 + 5 \beta_{1} - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{43} + ( -6 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} - 5 \beta_{5} ) q^{44} + ( -2 - \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{45} + ( 1 + \beta_{1} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{46} + ( 3 \beta_{1} - 6 \beta_{3} + 3 \beta_{4} ) q^{47} + ( 7 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} ) q^{48} + ( -5 - 4 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} ) q^{50} + ( -5 + 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{51} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{52} + ( 2 - \beta_{1} + 2 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} ) q^{53} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{54} + ( 1 - \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{55} + ( 1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{57} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{58} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{59} + ( -6 + 6 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{60} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{61} + ( 5 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{62} + ( -4 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{64} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{65} + ( -5 - 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{66} + ( 3 - \beta_{1} + 6 \beta_{2} + 8 \beta_{3} - \beta_{4} ) q^{67} + ( 4 - 5 \beta_{1} + 4 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{68} + ( 7 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{69} + ( 4 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{71} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} ) q^{72} + ( 5 - 4 \beta_{1} + 5 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{73} + ( -3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{74} + ( 3 + 4 \beta_{1} + \beta_{2} + 9 \beta_{4} - 4 \beta_{5} ) q^{75} + ( -6 + 4 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} ) q^{76} + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{78} + ( 4 - \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{79} + ( -7 \beta_{1} - 6 \beta_{4} + 7 \beta_{5} ) q^{80} + ( -5 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{81} + ( 1 - 6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + \beta_{4} + 6 \beta_{5} ) q^{82} + ( -1 + 2 \beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{83} + ( -5 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} ) q^{85} + ( 4 - 9 \beta_{1} + 4 \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{86} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} + ( 5 - 7 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} ) q^{88} + ( 1 + 6 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{89} + ( -11 - 6 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} ) q^{90} + ( -2 + 4 \beta_{1} - 2 \beta_{3} + 7 \beta_{4} ) q^{92} + ( -7 - \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{93} + ( -3 - 3 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} - 3 \beta_{4} ) q^{94} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{95} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{96} + ( 2 - 7 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{97} + ( -4 - 2 \beta_{1} - 2 \beta_{3} + 8 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 2q^{3} + 6q^{4} - 5q^{5} - q^{6} - 12q^{8} - 4q^{9} + O(q^{10}) \) \( 6q - 2q^{2} - 2q^{3} + 6q^{4} - 5q^{5} - q^{6} - 12q^{8} - 4q^{9} + 2q^{11} - 13q^{12} + 3q^{13} + 11q^{15} + 6q^{16} - 12q^{17} + 10q^{18} - 3q^{19} - 16q^{20} + 15q^{22} + 15q^{24} - 6q^{25} - q^{26} + 7q^{27} - q^{29} + 31q^{30} + 6q^{31} - 16q^{32} + 13q^{33} - 3q^{34} - 11q^{36} + 3q^{37} + 8q^{38} - 4q^{39} + 21q^{40} - 22q^{41} + 3q^{43} - 23q^{44} + q^{45} - 12q^{46} + 18q^{47} + 14q^{48} - 10q^{50} - 12q^{51} + 3q^{52} + 18q^{53} - 13q^{54} + 12q^{55} + 11q^{57} + 9q^{58} + 18q^{59} - 17q^{60} + 12q^{61} + 36q^{62} - 24q^{64} - 10q^{65} - 34q^{66} + 6q^{68} + 39q^{69} + 18q^{71} + 15q^{72} + 3q^{73} - 6q^{74} - 4q^{75} - 21q^{76} + 10q^{78} + 30q^{79} + 11q^{80} - 40q^{81} + 9q^{82} - 12q^{83} - 9q^{85} - 34q^{86} - 11q^{87} + 21q^{88} - 2q^{89} - 73q^{90} - 15q^{92} - 18q^{93} - 48q^{94} + 32q^{95} + 7q^{96} + 3q^{97} - 46q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(-1 - \beta_{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
−2.46050 −0.796790 + 1.53790i 4.05408 −1.29679 + 2.24611i 1.96050 3.78400i 0 −5.05408 −1.73025 2.45076i 3.19076 5.52655i
214.2 −0.239123 1.09097 + 1.34528i −1.94282 0.590972 1.02359i −0.260877 0.321688i 0 0.942820 −0.619562 + 2.93533i −0.141315 + 0.244765i
214.3 1.69963 −1.29418 1.15113i 0.888736 −1.79418 + 3.10761i −2.19963 1.95649i 0 −1.88874 0.349814 + 2.97954i −3.04944 + 5.28179i
373.1 −2.46050 −0.796790 1.53790i 4.05408 −1.29679 2.24611i 1.96050 + 3.78400i 0 −5.05408 −1.73025 + 2.45076i 3.19076 + 5.52655i
373.2 −0.239123 1.09097 1.34528i −1.94282 0.590972 + 1.02359i −0.260877 + 0.321688i 0 0.942820 −0.619562 2.93533i −0.141315 0.244765i
373.3 1.69963 −1.29418 + 1.15113i 0.888736 −1.79418 3.10761i −2.19963 + 1.95649i 0 −1.88874 0.349814 2.97954i −3.04944 5.28179i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.h.b 6
3.b odd 2 1 1323.2.h.e 6
7.b odd 2 1 441.2.h.c 6
7.c even 3 1 441.2.f.d 6
7.c even 3 1 441.2.g.d 6
7.d odd 6 1 63.2.f.b 6
7.d odd 6 1 441.2.g.e 6
9.c even 3 1 441.2.g.d 6
9.d odd 6 1 1323.2.g.b 6
21.c even 2 1 1323.2.h.d 6
21.g even 6 1 189.2.f.a 6
21.g even 6 1 1323.2.g.c 6
21.h odd 6 1 1323.2.f.c 6
21.h odd 6 1 1323.2.g.b 6
28.f even 6 1 1008.2.r.k 6
63.g even 3 1 441.2.f.d 6
63.h even 3 1 inner 441.2.h.b 6
63.h even 3 1 3969.2.a.m 3
63.i even 6 1 567.2.a.g 3
63.i even 6 1 1323.2.h.d 6
63.j odd 6 1 1323.2.h.e 6
63.j odd 6 1 3969.2.a.p 3
63.k odd 6 1 63.2.f.b 6
63.l odd 6 1 441.2.g.e 6
63.n odd 6 1 1323.2.f.c 6
63.o even 6 1 1323.2.g.c 6
63.s even 6 1 189.2.f.a 6
63.t odd 6 1 441.2.h.c 6
63.t odd 6 1 567.2.a.d 3
84.j odd 6 1 3024.2.r.g 6
252.n even 6 1 1008.2.r.k 6
252.r odd 6 1 9072.2.a.cd 3
252.bj even 6 1 9072.2.a.bq 3
252.bn odd 6 1 3024.2.r.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 7.d odd 6 1
63.2.f.b 6 63.k odd 6 1
189.2.f.a 6 21.g even 6 1
189.2.f.a 6 63.s even 6 1
441.2.f.d 6 7.c even 3 1
441.2.f.d 6 63.g even 3 1
441.2.g.d 6 7.c even 3 1
441.2.g.d 6 9.c even 3 1
441.2.g.e 6 7.d odd 6 1
441.2.g.e 6 63.l odd 6 1
441.2.h.b 6 1.a even 1 1 trivial
441.2.h.b 6 63.h even 3 1 inner
441.2.h.c 6 7.b odd 2 1
441.2.h.c 6 63.t odd 6 1
567.2.a.d 3 63.t odd 6 1
567.2.a.g 3 63.i even 6 1
1008.2.r.k 6 28.f even 6 1
1008.2.r.k 6 252.n even 6 1
1323.2.f.c 6 21.h odd 6 1
1323.2.f.c 6 63.n odd 6 1
1323.2.g.b 6 9.d odd 6 1
1323.2.g.b 6 21.h odd 6 1
1323.2.g.c 6 21.g even 6 1
1323.2.g.c 6 63.o even 6 1
1323.2.h.d 6 21.c even 2 1
1323.2.h.d 6 63.i even 6 1
1323.2.h.e 6 3.b odd 2 1
1323.2.h.e 6 63.j odd 6 1
3024.2.r.g 6 84.j odd 6 1
3024.2.r.g 6 252.bn odd 6 1
3969.2.a.m 3 63.h even 3 1
3969.2.a.p 3 63.j odd 6 1
9072.2.a.bq 3 252.bj even 6 1
9072.2.a.cd 3 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 1 \)
\( T_{5}^{6} + 5 T_{5}^{5} + 23 T_{5}^{4} + 32 T_{5}^{3} + 59 T_{5}^{2} - 22 T_{5} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} + 3 T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6} )^{2} \)
$3$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 18 T^{5} + 27 T^{6} \)
$5$ \( 1 + 5 T + 8 T^{2} + 7 T^{3} + 9 T^{4} - 62 T^{5} - 299 T^{6} - 310 T^{7} + 225 T^{8} + 875 T^{9} + 5000 T^{10} + 15625 T^{11} + 15625 T^{12} \)
$7$ 1
$11$ \( 1 - 2 T - 10 T^{2} - 34 T^{3} + 48 T^{4} + 416 T^{5} + 31 T^{6} + 4576 T^{7} + 5808 T^{8} - 45254 T^{9} - 146410 T^{10} - 322102 T^{11} + 1771561 T^{12} \)
$13$ \( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{3} \)
$17$ \( 1 + 12 T + 54 T^{2} + 210 T^{3} + 1350 T^{4} + 5898 T^{5} + 19735 T^{6} + 100266 T^{7} + 390150 T^{8} + 1031730 T^{9} + 4510134 T^{10} + 17038284 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 36 T^{2} - 18 T^{3} + 468 T^{4} + 324 T^{5} - 5393 T^{6} + 7452 T^{7} + 247572 T^{8} - 219006 T^{9} - 10074276 T^{10} + 148035889 T^{12} \)
$29$ \( 1 + T - 82 T^{2} - 31 T^{3} + 4425 T^{4} + 758 T^{5} - 148595 T^{6} + 21982 T^{7} + 3721425 T^{8} - 756059 T^{9} - 57997042 T^{10} + 20511149 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 - 3 T + 69 T^{2} - 213 T^{3} + 2139 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 322788 T^{7} + 316239 T^{8} + 22034055 T^{9} - 89959728 T^{10} - 208031871 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 5539756 T^{7} + 30984192 T^{8} + 124195642 T^{9} + 582106766 T^{10} + 2548836422 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 566310 T^{7} + 227427 T^{8} + 45239483 T^{9} - 184615254 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6} )^{2} \)
$53$ \( 1 - 18 T + 90 T^{2} - 378 T^{3} + 7848 T^{4} - 52668 T^{5} + 160459 T^{6} - 2791404 T^{7} + 22045032 T^{8} - 56275506 T^{9} + 710143290 T^{10} - 7527518874 T^{11} + 22164361129 T^{12} \)
$59$ \( ( 1 - 9 T + 171 T^{2} - 999 T^{3} + 10089 T^{4} - 31329 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 - 6 T + 162 T^{2} - 665 T^{3} + 9882 T^{4} - 22326 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( ( 1 - 6 T^{2} + 683 T^{3} - 402 T^{4} + 300763 T^{6} )^{2} \)
$71$ \( ( 1 - 9 T + 207 T^{2} - 1197 T^{3} + 14697 T^{4} - 45369 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 2071740 T^{7} - 16866285 T^{8} + 470321553 T^{9} - 1192726122 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \)
$79$ \( ( 1 - 15 T + 189 T^{2} - 1601 T^{3} + 14931 T^{4} - 93615 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 7506852 T^{7} + 239819868 T^{8} - 332780034 T^{9} - 6833998224 T^{10} + 47268487716 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 2 T - 112 T^{2} - 1238 T^{3} + 1662 T^{4} + 59806 T^{5} + 720895 T^{6} + 5322734 T^{7} + 13164702 T^{8} - 872751622 T^{9} - 7027130992 T^{10} + 11168118898 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 7614888 T^{7} + 133824207 T^{8} - 522961629 T^{9} - 14872919208 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \)
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